On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Applied Mathematical Modelling 36 (2012) 3411–3418

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Applied Mathematical Modelling journal homepage: www.elsevier.com/locate/apm

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions J. Martín-Vaquero a,⇑, B.A. Wade b a b

Departamento de Matemática Aplicada, E.T.S.I.I. Béjar, Universidad de Salamanca, E37700 Béjar, Salamanca, Spain Department of Mathematical Sciences, University of Wisconsin–Milwaukee, Milwaukee, WI 53201-0413, USA

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 December 2010 Received in revised form 9 October 2011 Accepted 16 October 2011 Available online 21 October 2011 Keywords: Klein–Gordon equations Nonlocal boundary conditions Numerical methods Wave equation

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the onedimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms. Ó 2011 Elsevier Inc. All rights reserved.

1. Introduction In the last few years, many physical phenomena have been formulated with non-local mathematical models. These physical phenomena are modeled by non-classical parabolic [1–8], elliptic [9], hyperbolic [10–12] or hyperbolic-parabolic [13] problems. The present work focuses on hyperbolic problems with non-local boundary value problems. The theoretical study of initial-boundary value problems in one space variable, which involve an integral over the spatial domain of a function of the desired solution that may appear in a boundary condition, is investigated in several articles [10,11,14–16]. In the last years, the development of numerical techniques for the solution of the hyperbolic nonlocal boundary value problems has been an important research topic in many branches of science and engineering. Particularly viscoelasticity has been the subject of some recent works [17,18]. Here, we consider the following hyperbolic problem with two non-local constraints (in place of the standard boundary conditions):

@2u @2u ¼ 2 þ qðx; tÞ; @x @t2

0 6 x 6 1;

0 6 t 6 tf ;

(where q(x, t) is a function sufficiently differentiable, tf 2 R+), with initial condition

⇑ Corresponding author. E-mail addresses: [email protected] (J. Martín-Vaquero), [email protected] (B.A. Wade). 0307-904X/$ - see front matter Ó 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.apm.2011.10.021

ð1Þ

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J. Martín-Vaquero, B.A. Wade / Applied Mathematical Modelling 36 (2012) 3411–3418

uðx; 0Þ ¼ f1 ðxÞ;

0 6 x 6 1;

ut ðx; 0Þ ¼ f2 ðxÞ;

ð2Þ

0 6 x 6 1;

ð3Þ

subject to the boundary conditions

uð0; tÞ ¼

Z

1

0

uð1; tÞ ¼

Z

/ðx; tÞuðx; tÞ dx þ g 1 ðtÞ;

0 6 t 6 tf ;

ð4Þ

wðx; tÞuðx; tÞ dx þ g 2 ðtÞ;

0 6 t 6 tf ;

ð5Þ

1

0

where q, f1, f2, g1, g2, / and w are known functions (we will assume enough smoothness in the functions to get the desire orders of convergency of the algorithms). We assume that these functions satisfy the conditions in order that the solution of (1)–(5) exists and is unique. Proofs of the existence, uniqueness, and continuous dependence for some particular cases of the model are studied in [10,17]. In [19], a numerical procedure for solving a similar problem by reducing it into a system of linear algebraic equations is presented. In [20], three computational techniques are compared. In this paper, we build a family of higher-order methods for a more general kind of problem and compare the new sixth-order method with the one found to be optimal in [20] for linear problems. We also adapt the fourth-order algorithm for nonlinear equations, in particular we shall show the efficiency of the new formula with the dimensionless form of the phi-four equation [21,22]. This is a particular case of the Klein–Gordon nonlinear equation that arises in theoretical physics:

@2u @2u ¼ 2 þ FðuÞ: @x @t2

ð6Þ

It represents the equation of motion of a quantum scalar or a pseudo-scalar field. If F(u) = sin (u), it is known as the Sine– Gordon equation which also arises in a large number of areas of physics, for example, crystal dislocation theory, self-induced transparency, laser physics and particle physics [23–25]; if F(u) = eu, (6) is the Liouville equation; if F(u) = sinh (u), gives the sinh-Gordon equation. 2. Three-level explicit methods In [20] three different schemes were proposed to solve a specific hyperbolic problem with non-local conditions. The author stated that the following so-called ‘‘optimal explicit technique’’

unþ1 ¼ i

rðr  1Þ n rð4  rÞ n ð4  rÞðr  1Þ n rð4  rÞ n rðr  1Þ n 2 þ k qni ; ui2 þ ui1 þ ui þ uiþ1 þ uiþ2  un1 i 12 3 2 3 12

ð7Þ

was fourth-order, but the leading terms of the error are 2

2

ð4  5r þ r 2 Þuk

þ

k ðuð4;0Þ  u0;4 Þ 12

ð8Þ

and the order depends on the value of r (r = k2/h2 = (Dt)2/(Dx)2; we are using the classic notation). Additionally, this technique requires four equations, except when r = 1 or 4 (when only two are necessary). If r = 1, both equations can be given through numerical approximations of the conditions (4) and (5) as we show below. The three procedures in [20] and the one that we shall derive at this paper are three-level techniques and, therefore, the solution on the initial time-step after is required. This is usually done through the Taylor expansion in time and Eq. (3); in this way, Dehghan gave [20]

u1i ¼ u0i þ kðut Þ0i þ

 2 k u0i1  2u0i þ u0iþ1 2h

2

þ qðxi ; kÞ:

ð9Þ

However, formula (9) is not accurate since if we apply Taylor series to the previous equation we obtain that the leading terms are 2

q þ

k q 3 þ Oðk Þ: 2

ð10Þ

There are better procedures for the first step, for example,

u1i

¼

u0i

þ

kðut Þ0i

þ

 2 k u0i1  2u0i þ u0iþ1 2h

2

2

þ

k qðxi ; 0Þ : 2

ð11Þ

2.1. The new family of higher-order methods The first purpose of this part of the paper is the construction of a family of methods for the linear Eq. (1). We will begin by deriving a fourth-order method, then we will vanish the leading term of the truncation error of this algorithm to obtain a

J. Martín-Vaquero, B.A. Wade / Applied Mathematical Modelling 36 (2012) 3411–3418

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sixth-order code, repeating this procedure it is possible to obtain algorithms with eight, tenth, . . ., order. Later, we will show how we can derive the methods for the nonlinear equations. Suppose that we have from (1) without q(x, t), the following formula:

unþ1 ¼ uni1 þ uniþ1  un1 ; i i which is exact for the homogeneous problem; therefore, it is possible to derive a family of higher-order finite difference schemes for the non-homogenous problem, canceling the Taylor expansions with the derivatives of q(x, t). For example, a fourth-order method for (1) is

unþ1 ¼ uni1 þ uniþ1  un1 þk i i

2

qni þ

! 2 k ðqxx Þni þ ðqtt Þni : 12

ð12Þ

This algorithm is derived by vanishing the two first terms of the truncation error when unþ1 ¼ uni1 þ uniþ1  un1 is applied to i i the non-homogenous problem (1), and now, the leading term of the truncation error is 4

k ðqtttt þ qttxx þ qxxxx Þ 5 þ Oðk Þ: 360

ð13Þ

Therefore, we can obtain the sixth-order method

unþ1 ¼ uni1 þ uniþ1  un1 þk i i

2

qni þ

!  2 4 k ðqxx Þni þ ðqtt Þni k ððqxxxx Þni þ ðqttxx Þni þ ðqtttt Þni Þ ; þ 360 12

ð14Þ

whose leading term of the truncation error is 6

k ðutttttttt  uxxxxxxxx Þ 7 þ Oðk Þ: 20160

ð15Þ

Again, we can also express this truncation error in terms of q (as in Eq. (13)) and vanishing it we would get an eight-order formula. In this way, it is possible to get algorithms of higher order. However, these kinds of formulae are not frequently given in terms of qxx or qtt, because it is possible to solve a more general problem if we are able to use qni ; qniþ1 ; qni1 instead. In this case, we can also solve efficiently the very interesting nonlinear equation

@2u @2u ¼ 2 þ qðx; t; uÞ; @x @t2

0 6 x 6 1;

0 6 t 6 tf ;

ð16Þ

for given q depending also on u. For example, we can build a fourth-order algorithm (for linear equations) if we replace k2 ððqxx Þni þðqtt Þni Þ qni þ in (12) by 12

qnþ1 þ qniþ1 þ qni1 þ qn1 2qn i i þ i 12 3

ð17Þ

and the leading term of the truncation error for equation

unþ1 i

¼

uni1

þ

uniþ1



un1 i

þk

2

qnþ1 þ qniþ1 þ qni1 þ qn1 2qn i i þ i 12 3

! ð18Þ

is then 4

k ð3qtttt þ 2qttxx  3qxxxx Þ 5 þ Oðk Þ: 720

ð19Þ

Obviously, it is possible to do something similar with the sixth- and eight-order formulas, but in this case, it would be necessary to evaluate q at many more points. To study the stability of the new family of higher-order methods we can proceed using the theoretical results in [26, Sections 8 and 9]. First of all, we shall reduce to consider only the homogeneous problem, for the inhomogeneous problem we merely remember that using Duhamel’s principle, a scheme is stable for the equation Pk,hv = f if it is stable for the equation Pk,hv = 0 as it is shown in [26, Section 9]. Now, we can use the following theorem: Theorem 1. [26, Theorem 8.2.1] If the amplification polynomial U(g, h) for a second-order time-dependent equation is explicitly independent of h and k, then the necessary and sufficient condition for the finite difference scheme to be stable is that all roots, gv(h), satisfy the following conditions: (a) jgv(h)j 6 1 and (b) if jgv(h)j = 1, then gv(h) must be at most a double root.

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Therefore, we can conclude the following result: Theorem 2. The family of higher-order methods derived in this section is stable. Proof. Using the Von Neumann analysis, if we consider the well-known Fourier inversion formula we obtain the amplification factor

g 2  gðeih þ eih Þ þ 1; whose roots are eih and eih and therefore jgv(h)j = 1 in both cases and they are simple roots, except when h = 0, p, when the root is double. h 2.2. Numerical approximation of the nonlocal conditions Eqs. (12), (14) or (18) present M  1 (for i = 1, . . . , M  1) linear equations in M + 1 (for i = 0, . . . , M) unknown variables for each n. In order to solve the linear system, we need two more equations, which we can obtain (as in [27]) by approximating the integrals (4) and (5) numerically by the fourth- or the sixth-order Simpson’s composite formulas. The fourth-order Simpson’s composite formula can be formulated in the following way:

unþ1 ¼ uð0; t nþ1 Þ ¼ 0 ¼

Z

1

/ðx; tnþ1 Þuðx; t nþ1 Þ dx þ g 1 ðtnþ1 Þ

0

M=2 M=21 X X nþ1 h nþ1 nþ1 nþ1 nþ1 /nþ1 þ4 /nþ1 /2j unþ1 0 u0 2j1 u2j1 þ 2 2j þ /M uM 3 j¼1 j¼1

and

unþ1 ¼ uð1; tnþ1 Þ ¼ M

Z

! 4

ð20Þ

4

ð21Þ

þ Oðh Þ þ g nþ1 1

1

wðx; t nþ1 Þuðx; tnþ1 Þ dx þ g 2 ðt nþ1 Þ

0

M=2 M=21 X X nþ1 h nþ1 nþ1 nþ1 nþ1 nþ1 wnþ1 u þ 4 w u þ 2 w2j unþ1 ¼ 0 2j1 2j1 0 2j þ wM uM 3 j¼1 j¼1

! þ g nþ1 þ Oðh Þ; 2

M being even here. Therefore, we have

 nþ1  nþ1 nþ1 h/0  3 unþ1 þ 4h/nþ1 þ    þ h/nþ1 ’ 3g nþ1 1 u1 M uM 0 1

ð22Þ

  nþ1 nþ1 nþ1 hwnþ1 þ 4hwnþ1 þ    þ hwnþ1 ’ 3g nþ1 0 u0 1 u1 M  3 uM 2 :

ð23Þ

and

Accordingly, we can write the two linear equations that we need as:

    nþ1 unþ1 ¼ Y 1 W 1 hwnþ1 M  3  W 2 h/M 0

ð24Þ

    unþ1 ¼ Y 1 W 2 h/nþ1  3  W 1 h/nþ1 ; 0 M M

ð25Þ

and

where

W 1 ¼ 2h

M=2 X

! nþ1 /nþ1 2i1 u2i1

 4h

i¼1

W 2 ¼ 2h

M=2 X

M=21 X

! nþ1 /nþ1 2i u2i

 3g nþ1 1 ;

ð26Þ

 3g nþ1 2

ð27Þ

i¼1

! nþ1 wnþ1 2i1 u2i1

i¼1

 4h

M=21 X

! nþ1 wnþ1 2i u2i

i¼1

and

     2 nþ1 nþ1 Y ¼ h/nþ1  3 hwnþ1 ¼ 9  3h /nþ1 þ wnþ1 : 0 M  3  h /0 wM 0 M

ð28Þ

We can also derive a sixth-order formula to approximate the integrals:

unþ1 ¼ uð0; t nþ1 Þ ¼ 0

Z

1

/ðx; tnþ1 Þuðx; t nþ1 Þ dx þ g 1 ðtnþ1 Þ

0

M=2 M=41 M=42 X X nþ1 X nþ1 2h nþ1 nþ1 nþ1 nþ1 7/nþ1 þ 32 /nþ1 /4jþ2 unþ1 /4jþ4 unþ1 ¼ 0 u0 2j1 u2j1 þ 12 4jþ2 þ 14 4jþ4 þ 7/M uM 45 j¼1 j¼0 j¼0

! 6

þ g nþ1 þ Oðh Þ 1

ð29Þ

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J. Martín-Vaquero, B.A. Wade / Applied Mathematical Modelling 36 (2012) 3411–3418

and

unþ1 ¼ uð1; tnþ1 Þ ¼ M

Z

1

wðx; t nþ1 Þuðx; tnþ1 Þ dx þ g 2 ðtnþ1 Þ

0

M=2 M=41 M=42 X X nþ1 X nþ1 2h nþ1 nþ1 nþ1 nþ1 nþ1 nþ1 7wnþ1 u þ 32 w u þ 12 w u þ 14 w4jþ4 unþ1 ¼ 0 2j1 2j1 4jþ2 4jþ2 0 4jþ4 þ 7wM uM 45 j¼1 j¼0 j¼0

! 6

þ Oðh Þ; þ g nþ1 2

ð30Þ

in this case, we need M being a multiple of 4. Hence, we can write the two linear relations that we need as:

    nþ1 unþ1 ¼ Y 1 W 6;1 14hwnþ1 M  45  W 6;2 14h/M 6 0

ð31Þ

    unþ1 ¼ Y 1 W 6;2 14h/nþ1  45  W 6;1 14hwnþ1 ; 0 M M 6

ð32Þ

and

where

W 6;1 ¼ 64h

M=2 X

! nþ1 /nþ1 2i1 u2i1

 24h

i¼1

W 6;2 ¼ 64h

M=41 X

! nþ1 /nþ1 4iþ2 u4iþ2

 28

i¼0

M=2 X

! nþ1 wnþ1 2i1 u2i1

 24h

i¼1

M=41 X

M=42 X

! nþ1 /nþ1 4iþ4 u4iþ4

 45g nþ1 1 ;

ð33Þ

 45g nþ1 2

ð34Þ

i¼0

! nþ1 wnþ1 4iþ2 u4iþ2

i¼0

 28

M=42 X

! nþ1 wnþ1 4iþ4 u4iþ4

i¼0

and

   2 nþ1 nþ1 nþ1 Y 6 ¼ 2025  630h /nþ1 þ wnþ1 þ 196h /nþ1 : 0 M 0 wM  /M þ w0

ð35Þ

3. Numerical examples In this section we give some results of numerical experiments with the methods shown on the preceding sections and compare with algorithms proposed in [20], to support our theoretical discussion. Since the principal goal is showing the efficiency of the new methods, we have considered that the solution in the first step, u(k, xi) is exact for both methods for a better comparison of the algorithms. Test 1. We will begin by considering the simple test problem (1)–(5) with

g 1 ðtÞ ¼ ð2 þ eÞet1 t; g 2 ðtÞ ¼ g 1 ðtÞ; /ðx; tÞ ¼ 1;

0 < t < 1;

0 < t < 1;

0 < x < 1;

wðx; tÞ ¼ 1  x;

0 < x < 1;

qðx; tÞ ¼ 2etx ðt  xÞ;

0 6 t 6 1;

0
and with solution

uðx; tÞ ¼ xtetx : In Fig. 1 we compare the step size employed with the maximum error at t 2 [0, 1], that is, maxi=1,. . .,M;j=1,. . .,NjU(ih, jk)aprox  U(ih, jk)exactj using the ‘‘optimal explicit technique’’ in [20] and the sixth-order algorithm that we derived in the previous section. In Fig. 2 we compare the CPU time employed and the maximum error at t 2 [0, 1] (maxi=1,. . .,M;j=1,. . .,NjU(i h, j k)aprox  U(i h, jk)exactj) using the same methods. We see clearly that the new method gets better results. Test 2. In this case, we choose the Klein–Gordon nonlinear equation with F(u) = u  u3. This is the dimensionless form of the phi-four equation [21,22,28]. This equation arises in quantum field theory in the study of quartic interaction theory and its 1-soliton solution is given by

uðx; tÞ ¼

A ; cosh k

pffiffiffi 1 ffi where A ¼ 2; k ¼ Bðx  ltÞ; B ¼ pffiffiffiffiffiffiffiffi . Here, A is the amplitude of the soliton, B is the inverse width of the soliton and m is l2 1 its velocity. In this case, we have the exact solution so we can compare the numerical results, however it also can have perturbation terms.

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J. Martín-Vaquero, B.A. Wade / Applied Mathematical Modelling 36 (2012) 3411–3418

10

-3

10

-5

10

-7

10

-9

Sixth-order

Error

"Optimal explicit"

10

-11

10

-13

.2

.1

.05

.02

.01

.005

Length step Fig. 1. Numerical comparison of the step size employed versus the numerical errors in Test 1. The ‘‘optimal explicit technique’’ is the algorithm proposed in [20].

10

-3

Sixth-order -5

10

-7

10

-9

"Optimal explicit"

Error

10

10

-11

10

-13

.01

.1

1

10

CPU Time Fig. 2. Numerical comparison of the CPU time (s) versus the numerical errors in Test 1. The ‘‘optimal explicit technique’’ is the algorithm proposed in [20].

We have solved

utt  uxx  u þ u3 ¼ 0;

0 6 x 6 1;

06t61

ð36Þ

with the following boundary conditions

uð0; tÞ ¼ g 1 ðtÞ;

0 6 t 6 1;

ð37Þ

uð1; tÞ ¼ g 2 ðtÞ;

0 6 t 6 1;

ð38Þ

g1(t) and g2(t) being such that

pffiffiffi 2   uðx; tÞ ¼ pffiffi cosh x2t 3 is the solution of the problem. In Fig. 3 we compare the CPU time employed with the maximum error at t 2 [0, 1], that is, maxi=1,. . .,M;j=1,. . .,NjU(ih, jk)aprox  U(ih, jk)exactj using the ‘‘optimal explicit technique’’ in [20] and the fourth-order algorithm that we derived in the previous section.

J. Martín-Vaquero, B.A. Wade / Applied Mathematical Modelling 36 (2012) 3411–3418

3417

Fourth-order formula

Numerical error

10

10

10

10

-3

"Opimtal explicit"

-5

-7

-9

.01

.1

1

10

CPU Time (s) Fig. 3. Numerical comparison of the CPU time (s) employed versus the numerical error at t 2 [0, 1], in Test 2. The ‘‘optimal explicit technique’’ is the algorithm proposed in [20].

Fourth-order formula

Numerical errors

10 -4

"Optimal explicit"

10 -6

10 -8

10 -10 .01

.1

1

10

CPU Time (s) Fig. 4. Numerical comparison of the CPU time (s) employed versus the numerical errors at t = 1 in Test 2. The ‘‘optimal explicit technique’’ is the algorithm proposed in [20].

In Fig. 4 we compare the CPU time employed and the maximum error at t = 1 (maxi=1,. . .,MjU(ih, 1)aprox  U(ih, 1)exactj) using the same methods. We see that the new method yields better results. 4. Conclusions In this paper, we derive a new family of high-order stable three-level algorithms to solve the wave equation, if we also use high-order formulae to approximate integral conditions, then we can solve efficiently the hyperbolic equation with non-local boundary conditions. We detail the proofs of the stability and the consistency of these schemes. Additionally, these formulae can be adapted to nonlinear equations, in particular, we used the fourth-order scheme to approximate the nonlinear Klein–Gordon equations with good numerical results, as we showed, comparing with other existing methods. Acknowledgments This work has been supported by Grant FS13-2008 from Fundación Memoria de D. Samuel Solórzano Barruso. The second author’s research has been supported in part by the US National Security Agency under Grant Agreement Number H98230-

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